## Linear Operators: Spectral theory |

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Page 891

scalar function f with respect to the operator valued set function E. In the present

chapter we shall only integrate bounded ... We shall use the term 2-

...

scalar function f with respect to the operator valued set function E. In the present

chapter we shall only integrate bounded ... We shall use the term 2-

**simple****function**for any function on S having the form n (viii) f = X xix, i-1 where zo, is the...

Page 892

This shows that if the sequence {fi} of 2-

then the sequence {ssf,(s)|E(ds)} of integrals converges, and its limit depends

only upon f and not upon the particular sequence {fi} used to approximate f.

This shows that if the sequence {fi} of 2-

**simple functions**converges in B(S, 2) to fthen the sequence {ssf,(s)|E(ds)} of integrals converges, and its limit depends

only upon f and not upon the particular sequence {fi} used to approximate f.

Page 1218

The following lemma is, in the case of scalar functions, a well known theorem of

Lusin. 17 LEMMA. Let u be a ... Since u(R) < oo every u-measurable function f is

the limit in u-measure of a sequence {f,} of u-

...

The following lemma is, in the case of scalar functions, a well known theorem of

Lusin. 17 LEMMA. Let u be a ... Since u(R) < oo every u-measurable function f is

the limit in u-measure of a sequence {f,} of u-

**simple functions**. In view of Corollary...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 other sections not shown

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additive Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero